# Math Help - Question related to Weierstrass criteria for uniformly convergent series of functions

1. ## Question related to Weierstrass criteria for uniformly convergent series of functions

hello to all

I have some issue of understanding this criteria... it says:

Let

$\displaystyle (a_n)_{n\in \mathbb{N}$

be positive numerical sequence for which is worth that for almost every

$\displaystyle n \in \mathbb{N}$

and every

$\displaystyle x\in A$

is satisfied that

$\displaystyle |f_n (x)|\le a_n$

then if series:

$\displaystyle \sum_{n=1} ^{+\infty} a_n$

is convergent then series (of functions) :

$\displaystyle \sum_{n=1} ^{+\infty} f_n(x)$

is uniformly convergent on set A.
my question is why it is "uniformly" convergent ? I just can't see the fact why we have here uniformly convergent series...

Thanks for any help

2. The Wikipedia article on it explains it nicely: Uniform convergence - Wikipedia, the free encyclopedia

3. I don't have a problem with understanding the meaning of uniform convergence... I have problem with this criteria... perhaps i was not clear about it very good... I'm also having problem with English hehehe

It's like this.... Why with assumption that (a_n) is sequence of positive numbers... and if ... (everything in the criteria) .... how someone can conclude that series of functions uniformly converge ... because of what ? Is it because of absolutely convergence of numerical series... ? or what ? how do I know that It's uniformly converge and not just by points...